Tutorial Track

D. Asperó: Side conditions and (iterated) forcing

I am planning to give a slowly paced introduction to the construction of forcing notions incorporating suitable systems of models as side conditions. Some of these forcing notions will be naturally built as limits of forcing iterations and some will not. The main focus will be on the applications of the method.

J. Bagaria: An Introduction to Hyperstationary Sets

We shall give an introduction to the theory of hyperstationary sets. Hyperstationarity generalizes the notion of stationary set (i.e., 1-stationarity) and stationary reflection (i.e., 2-stationarity). It turns out that $\xi$-simultaneous-stationary ordinals are precisely the non-isolated points in the $\xi$-th topology on the ordinal numbers, which is obtained by refining the usual interval topology using the Cantor's derivative operator and iterating the refining process $\xi$-many steps. Thus, we will discuss the connections between $\xi$-stationarity, $\xi$-simultaneous-stationarity, derived topologies on the ordinals, and the ideals of non-$\xi$-stationary sets and their dual filters. We will show that in the constructible universe $L$ a regular cardinal is $(\xi +1)$-stationary if and only if it is $\Pi^1_\xi$-indescribable, and we will present some very recent results about the consistency strength of $\xi$-stationarity.

C. Brech: Generalizing Schreier families to large index sets

Products of the Schreier family have been used in Banach space theory to built important objects such as the Tsirelson space. In order to generalize these constructions to the nonseparable setting, the families had to be generalized to uncountable index sets. The Schreier family is hereditary, compact and spreading but when passing to the uncountable level, we cannot expect them to be spreading if we want to keep compactness, for example.

We will briefly present the link between the families and their corresponding Banach spaces and will focus on how to define and construct those families. The right definition of a product of a family on a large index set by the Schreier family is crucial and there is a weaker and a stronger version of it in terms of the corresponding Cantor-Bendixson index. After introducing these notions, we will present the method (appearing in our recent joint work with Jordi Lopez-Abad and Stevo Todorcevic) to construct such families below the first Mahlo cardinal. It involves defining a family on a tree out of a family on its chains and a family on its antichains, and analyzing the combinatorial structure of it using Ramsey methods.

A. Marks: Geometrical paradoxes and descriptive set theory

The well-known Banach–Tarski paradox states that the unit ball in $\mathbf R^3$ can be partitioned into finitely many pieces that can be rearranged by rotations and translations to form two unit balls. The study of descriptive-set-theoretic aspects of this paradox has a long history. For example, in 1930, Marczewski asked whether the pieces used in this equidecomposition can have the Baire property. Dougherty and Foreman gave a positive answer to this question in 1994.

Recent progress on matching theorems in the field of descriptive graph combinatorics has shed new light on the general question of when the pieces used in equidecompositions such as the Banach–Tarski paradox may possess various regularity properties such as Lebesgue measurability or the Baire property. We will discuss some of these developments, and applications to the existence of finitely additive invariant measures.